10 INTEGRALS OF VECTOR-VALUED FUNCTIONS 155
and «// are integrable, and we have
i (
from which it follows immediately that
The set JS?c(X, A*) of /-i-integrable functions on X with values in C is a1 vector
space over C, and/W fdfj, is a linear form on this vector space. Further-
more, if/is any complex-valued integrable function, then |/| isintegrable and
(13.10.3)
sl
I/I
For since |/| = ((^/)2 + C//)2)1/2, it follows that |/| is measurable
(13.9.6), and we have M*(l/l) ^ M*(l*/l) + /**(I>/D- Hence the first assertion,
by (13.9.13). Also there exists a complex number C of absolute value 1 such
that CK/) = IX/)I, and hence |/i(/)| = *WC/)) = M^(C/)) £ Kl/l), which
establishes (13,10.3).
If E' is the dual of the finite-dimensional vector space E, then to say that
are integrable is equivalent to saying that, for each *' e E', the mapping
><f(.x),z'> is integrable (because it is a linear combination of the jQ. We
generalize this as follows: if I is any set, then a mapping x\-*fx of X into the
vector space K! (where K = R or C) is scalarly ^.-integrable if for each
a e I the function x\->fx(a) is integrable.
(13.10.4) LetEbeaFrdchetspace, E' its dual ([IAS). Let x^^ fx be a scalarly
integrable mapping of X into E' such that, for each convergent sequence (art)
in E, there exists a function g ^ 0 defined on X such that n*(g) < + oo and
lfx(an)l ^ ff(x)for all n, almost everywhere in X. Then there exists a continuous
linear form z' e E' on E such that <z, z'> = J fx(z) d^(x)for all t e E.
By (3.13.4) it is enough to show that if (aw) is a sequence in E with limit z
then f fJXz) dn(x)= lim f f,(an) ^u(jc); and this follows from (13.8.4).
J M-'-OO*'
The linear form z' so defined is called the integral (or weak integral) of
the function x^>fx with respect to /^, and is denoted by | fx d^(x). Hence, forsets, the number